Let $A$ be a ring, $I\subset A$ an ideal, $M$, $N$ $A$-modules such that $IM=0$ and $IN=0$. Then the modules extend to $A/I$-modules, and we have
$$\operatorname{Hom}_A(M,N)=\operatorname{Hom}_{A/I}(M,N).$$
But is this true for higher homological dimensions? i.e., is
$$\operatorname{Ext}_A^i(M,N)=\operatorname{Ext}^i_{A/I}(M,N)$$
true for all $i\geq0$?